To solve the equation

[ \log \left( \frac{10 \sqrt[3]{v}}{y \sqrt[4]{100}} \right) = 1 + \frac{1}{3} \log(v) - \log(y) - \frac{1}{2} ]

let’s break down each side and simplify the expression.

Step 1: Simplify the Left Side

The left side is

[ \log \left( \frac{10 \sqrt[3]{v}}{y \sqrt[4]{100}} \right) ]

Using the property of logarithms $\log \left( \frac{a}{b} \right) = \log(a) - \log(b)$, we can separate the terms:

[ \log \left( \frac{10 \sqrt[3]{v}}{y \sqrt[4]{100}} \right) = \log(10 \sqrt[3]{v}) - \log(y \sqrt[4]{100}) ]

Breaking Down $\log(10 \sqrt[3]{v})$:

1. Use the property $\log(ab) = \log(a) + \log(b)$: [ \log(10 \sqrt[3]{v}) = \log(10) + \log(\sqrt[3]{v}) ]

2. Simplify further using $\log(10) = 1$ and $\log(\sqrt[3]{v}) = \frac{1}{3} \log(v)$: $\log(10 \sqrt[3]{v}) = 1 + \frac{1}{3} \log(v)$

Breaking Down $\log(y \sqrt[4]{100})$:

1. Again, using $\log(ab) = \log(a) + \log(b)$: [ \log(y \sqrt[4]{100}) = \log(y) + \log(\sqrt[4]{100}) ]

2. Since $\sqrt[4]{100} = 10^{1/2} = \sqrt{10}$, we find: $\log(\sqrt[4]{100}) = \frac{1}{2}$

3. Thus, $\log(y \sqrt[4]{100}) = \log(y) + \frac{1}{2}$

Putting it All Together

Substitute these simplified expressions back:

[ \log \left( \frac{10 \sqrt[3]{v}}{y \sqrt[4]{100}} \right) = \left( 1 + \frac{1}{3} \log(v) \right) - \left( \log(y) + \frac{1}{2} \right) ]

Expanding this, we get:

[ \log \left( \frac{10 \sqrt[3]{v}}{y \sqrt[4]{100}} \right) = 1 + \frac{1}{3} \log(v) - \log(y) - \frac{1}{2} ]

Final Result

This matches the right side of the original equation, so the equation holds:

[ \log \left( \frac{10 \sqrt[3]{v}}{y \sqrt[4]{100}} \right) = 1 + \frac{1}{3} \log(v) - \log(y) - \frac{1}{2} ]

Would you like further details, or have any questions?

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Here are 5 questions to deepen understanding:

1. What are some other logarithmic properties used to simplify complex expressions?
2. How can we use the chain rule of logarithms for nested roots?
3. Why is $\log(10) = 1$ in this context, and how would it change in other bases?
4. How would this approach change if we had variables instead of constants like 10 or 100?
5. What are alternative methods to validate logarithmic expressions?

Tip: To simplify logarithmic expressions, always break down complex terms using basic properties step-by-step.